7.5 Combination inside a tensor-product theory

When the selected sketch doctrine admits the required tensor product, its semantics makes the fixed space of combination explicit. Let \(\mathbb A\) and \(\mathbb B\) present two registered families of operations. A model of \(\mathbb A\otimes \mathbb B\) then supplies both families and the specified interchange squares that determine how their composites are compared. Combinational creativity searches for a new term, diagram, policy, adapter, or mechanism inside this presented double theory; it does not yet alter either factor or the interchange law.

This separates two superficially similar outcomes. Discovering a useful composite already generated by \(\mathbb A\otimes \mathbb B\) is combinational. A proof, exhaustive finite analysis, or other registered nonrealizability argument showing that no model in the selected semantic class can realize the observed interaction without a mixed generator or revised interchange equation motivates a transformational proposal. Failure of a finite search does not. The distinction is semantic rather than textual: a long composite may remain inside the fixed theory, whereas one small new generator may change its model category.

Finite composability is still an admission issue. A novel local pairing may reduce the interchange defect at one state but fail when repeated, transported, or composed around a loop. Accordingly, combinational DIAL experiments should test whether the proposed composite integrates into a stable finite operation, not only whether its first-order interaction is favorable.

. Call a DIAL result combinational only when the tensor factors and interchange presentation remain fixed, the constructed composite is novel relative to the registered repertoire, and its benefit persists under a finite composition or transport test. A required edit to either factor or to interchange begins a transformational trace.